Limit Cycles and Bifurcations in Two Dimensions
Stub. Isolated periodic orbits (limit cycles) — ruling them out, the Poincaré-Bendixson theorem guaranteeing them, Liénard systems, relaxation oscillations, and weakly nonlinear oscillators — then how 2D phase portraits bifurcate as a parameter varies: the 2D saddle-node/transcritical/pitchfork cases, the Hopf bifurcation (a fixed point spawning a limit cycle), oscillating chemical reactions, global bifurcations of cycles, hysteresis, coupled oscillators/quasiperiodicity, and Poincaré maps.
Corresponds to Strogatz, Chapters 7–8. A limit cycle is an isolated closed orbit — trajectories nearby spiral toward (or away from) it, unlike the whole continuum of closed orbits a conservative system has. Covers the standard techniques for ruling closed orbits out (Bendixson’s criterion) or in (the Poincaré-Bendixson theorem, a 2D-specific existence result with no higher-dimensional analogue — one reason chaos needs 3+ dimensions), Liénard systems and relaxation oscillations as concrete sources of limit cycles, and perturbative analysis of weakly nonlinear oscillators. The bifurcation theory of the earlier note extends to 2D with a genuinely new case: the Hopf bifurcation, where a stable fixed point turns unstable and sheds a small limit cycle — the generic mechanism by which oscillation switches on in a 2D system, worked out alongside chemical-reaction oscillators, global (as opposed to local) bifurcations of cycles, hysteresis, and quasiperiodicity in coupled oscillators. Poincaré maps (sampling a flow once per cycle) are introduced as the tool that reduces questions about limit cycles and their bifurcations to the discrete-map analysis of the next note.
Outline (TODO — flesh out each)
- Limit cycles — examples
- Ruling out closed orbits
- The Poincaré-Bendixson theorem
- Liénard systems
- Relaxation oscillations
- Weakly nonlinear oscillators
- Saddle-node, transcritical, and pitchfork bifurcations (2D)
- Hopf bifurcations
- Oscillating chemical reactions
- Global bifurcations of cycles
- Hysteresis in the driven pendulum and Josephson junction
- Coupled oscillators and quasiperiodicity
- Poincaré maps